Cancellation-free circuits: An approach for proving superlinear lower bounds for linear Boolean operators

Abstract

We continue to study the notion of cancellation-free linear circuits. We show that every matrix can be computed by a cancellationfree circuit, and almost all of these are at most a constant factor larger than the optimum linear circuit that computes the matrix. It appears to be easier to prove statements about the structure of cancellation-free linear circuits than for linear circuits in general. We prove two nontrivial superlinear lower bounds. We show that a cancellation-free linear circuit computing the n × n Sierpinski gasket matrix must use at least 1 2 n log n gates, and that this is tight. This supports a conjecture by Aaronson. Furthermore we show that a proof strategy for proving lower bounds on monotone circuits can be almost directly converted to prove lower bounds on cancellation-free linear circuits. We use this together with a result from extremal graph theory due to Andreev to prove a lower bound of Ω(n 2−ǫ ) for infinitely many n × n matrices for every ǫ > 0 for. These lower bounds for concrete matrices are almost optimal since all matrices can be computed with O n 2 log n gates.

On additive complexity of a sequence of matrices

“….k] ∪ {n}) ⊂ P , to the sum (1). At last, if [1..p] ∩ Q = ∅, then the output correctly computes a sum y Q ∈ p, q, ∅, [1.…”

“…1.4. Add the sums computed on the step 1.3 to sums (1). Complexity of this addition is the number of sums (1), i.e.…”

“…The proof of the lemma is similar to the proof of Th. 4 in [1]. Consider an arbitrary additive circuit Ψ implementing B p,q,n .…”

“…схемами из функциональных элементов сложения над группой (Z, +). 1 Базовые понятия схем и сложности см. в [1,4].…”

“…1 Базовые понятия схем и сложности см. в [1,4]. Сложность вычисления матрицы A аддитивными схемами над (Z, +) обозначим через L(A).…”